Download BIOEQ4: Bioequivalence Macro to Create both Table and SAS data set according to the FDA Bioequivalence Guidelines Issued in 1992

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BIOEQ4: BIOEQUIVALENCE MACRO TO CREATE BOTH TABLE AND SAS-DATA- SET
ACCORDING TO THE FDA BIOEQUIVALENCE GUIDELINES ISSUED IN 1992~
Mary Kay McPherson, Wyeth-Ayerst Research
Introduction:
In the pharmaceutical industry bioequivalence
analysis is required to prove that a given drug (generic or
other) fonnulation has a similar distribution in the body as
the original used in research. 13ioequivalence analysis can
be done by the company filing a new drug application in
order to replace the trial (or reference) fonnulation with a
marketed (or test) fonnulation. or by a generic company
trying to market a generic fonnulation (test) of the patented
(reference) drug. The Food and Drug Administration
(FDA) has developed strict guidelines to ensure equivalent
distribution of drugs in humans after administration.
Bioequivalence in humans is usually demonstrated
using pharmacokinetic (PK) parameters which look at how
drugs are distributed into and eliminated from the body.
The PK parameters.
AUc;. and t",... are estimated by
looking at blood. plasma. or serum drug concentrations
within a person over time. Bioequivalence analysis can
focus on the population. the individual or the average
distribution of drug in various subjects. This paper focuses
on average bioequivalence and the subsequently required
analysis.
c.....
Companies must prove average bioequivalence
using the FDA guidelines before a New Drug Application
(NDA) approvals can be made. The FDA guidelines have
evolved over time in order to stay abreast of current
statistical and pharmacokinetic methodologies yet provide
tangible rules to guide companies in demonstrating
bioequivalence. Older guidelines focus on arithmetic
means. i =!.
is implemented and requires a different specification of
the statistical model. The BIOEQ4 macro is capable of
managing any design comparing two formulations or
treatments and can be easily altered to handle studies that
attempt to prove bioequivalence among several
fonnulations.
The FDA requires companies use an ESTIMAlE
statement l • which takes into consideration dependencies
when estimating a standard error, to find the correct
estimates of the mean difference and corresponding
standard errors of the PK parameters.
Since the
ESTlMAlE statement is the only way in PROC GLM (in
SAS version 6.09) to include covariances when estimating
the error of difference in least square means and there is no
easy way to output and handle the results from an
ESTIMAlE statement. automating a bioequivalence
analysis is virtually impossible.
A key component involved in the automation of
bioequivalence analysis is the consideration of the
covariance structure. By using the LSMEANS statement
and options to output the variance-covariance matrix and
using the following equation:
one can find the variance of the difference in means which
corresponds to the results of the FDA-required
ESTlMAlE statement. The following estimate statement
will calculate the LS mean difference (test - reference)
where the fonnulations are ordered as reference first and
test second in CLASS page of PROC GLM and also when
the data is sorted using PROC SORT.
ESTIMATE
Ex,. while the newer 1992 guidelines focus
'TEST -REFERENCE' TRT -1 1;
" ,.J
The BIOEQ4 macro looks
Formats may affect this ordering which is crucial to the
correct results when BIOEQ4 is used.
at the geometric mean of the ratios of an individual' s test
and reference PK parameters as well as the ratio of the
arithmetic means. Various statistical inferences and tests
are then calculated based on these two forms.
The variance/covariance matrix can be estimated by using
the following equation:
on the geometric mean.
(iIx,);.
,.J
'
Variance Covariance = (X/X)-I~
Statistical Background:
Var(XR)
Bioequivalence is usually demonstrated by
implementing a basic 2 x 2 crossover where one group of
subjects receives fonnulation A followed by washout time
and then receives fonnulation B. The second group of
subjects receives the fonnulations in reverse order.
Occasionally a more intricate design than a 2 x 2 crossover
[
= Cov(X
T
•
X )
R
Cov(XR ~ X T)]
Var(XT)
where X is the design matrix and 0 2 is the estimate of the
true error variability. A cross section of the data created
by the LSMEANS statement has the following structure:
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-
NAME
-
LVAR1
LVAR1
FORM
A
B
• Xn )11.
_ (In II
)1'. -exp
;=1 XlU
COV2
COV~
0.00216504
0.00000173
0.00000173
0.00216504
.J1
= exl'\
=
Sample part of data set output by LSMEANS statement
within PROC GIM.
The variables COV1 and COV2 correspond to the 2 by 2
variance-covariance matrix
COVl
COV2
0.00216504 0.00000173
[ 0.00000173 0.00216504
where
By applying the data to the equation:
the estimate of the Variance of the difference in LS means
can be estimated as follows:
In
"X)
11
..2!.
n
i=1 xm
.J I" x )
exl'\ - !)n..2!.
n
I-I
XR1
1 •
(InXn -lnXRi)
n ,eJ
1 •
1•
InXn - InXRi)
n i-I
n 1=1
=
ex~
=
ex~
=
ex~ iT-iR )
E
E
E
i
is the mean using log.-transformed data,
Xn is the value of the test measurement of the ith
subject, and
XRi is the value of the reference measurement of
the ilh subject.
Likewise one can also find the estimate of the standard
error and confidence limits of the ratios simply by
exponentiating the corresponding standard error and
confidence limits of the difference in means obtained using
log..transformed data:
Involved in the analysis are the estimates based on
the ratio (test/reference) of geometric means, the ratio of the
arithmetic means and the difference of the arithmetic means.
the difference of the means can be found simply by using
the LSMEANS statement to find the formulation means and
then calculating the difference. The confidence intervals
can then be calculated accordingly using the above
estimates, as follows:
exp ( iR-iT ±
tr<,df
* JVar(iR-iT) )
Power associated with the F-test testing for
differences in formulation is calculated assuming that one
is either interested in a difference (test-reference) no larger
than Iog(1.2) for the log-transformed data or 0.2 *
reference mean for untransformed data. The noncentrality
parameter can be calculated as follows:
NC=
52
Var(iT-iR)
The geometric mean, of the ratios can be found by
using the log.-transformed data in the GLM procedure to
calculate an arithmetic mean (or average) from the
LSMEANS statement and subsequently exponentiating the
result The following equation may clarify the process. A
geometric mean of the ratios of test to reference could be
expressed as:
where
o
= log(1.2) for log.-transformed data,
= 0.2*reference mean for non-transformed data.,
Var(iT-iR)
=variance of the difference of the means.
A situation where the power reaches 80% is often referred
to as the ±80/20 rule since bioequi valence is declared if
the test mean falls within 20% of the reference mean with
80% power.
The power calculation may be slightly different
from those of other programs since it includes covariances
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that may possibly exit in the estimate of the treatment mean
difference standard error. Many noncentrality parameter
estimates are simplified by using either of the following
equations:
NC-(
0
(1*
)
2
the GLM procedure. FORMULA is the variable that
defines the bioequivalence comparitors. The variables
SEQUENCE, PERIOD and SUBJECT are common
statistical effects used in a crossover Analysis of Variance
(ANOVA).
=
..{iJn
or
BIOEQ4 can be invoked by using the following
NC=( 0
code:
which do not include covariance structures in the estimate
of the difference of the LS means. Thus, the power estimate
may be slightly different than in a simple software package,
though it is a more accurate estimate.
Macro:
The macro is set up in several sections: data
preparation, statistical analysis, bioequivalence analysis,
and table creation. Various amenities such as by-variable
processing, printing/no-printing, and table titling and
footnoting are concurrently programmed within the
sections. The macro can easily be altered to suit other
situations if there is adequate understanding of the GLM
procedure and its output.
%BIOEQ4(
%* input data set;
DRUGXPPP,
%* output data set;
BIOEQ,
%* PK variables;
PK1 PK2 PK3,
3,
%* number of PK vars;
DOSE,
%* by-variable;
%STR('MONT'), %* Reference formulation;
%STR('WWPI'),
%* Test formulation;
SUB (SEQ) SEQ DAY TREAT,
%* stat model;
SUB SEQ DAY TREAT, %* Class variables;
%* Formulation var;
TREAT,
TITLE1,
TITLE2,
TITLE3,
%STR("CMAX" "TMAX" "AUCT"), %*Labels;
FOOTNOTE 1 ,
FOOTNOTE2,
%* Print results and output;
PRINT,
) ;
The data can be entered into SAS and be structured
so that each subject's PK parameters for one formulation
and all by-processing variables be located within one
record, as shown below:
S
F
U
0
B
J
E
C
D
0
S
E
T
1
1
2
2
10
10
10
10
R
M
U
S
E
Q
U
E
L
A
N
C
A
B
A
B
1
1
2
2
P
E
R
I
0
C
M
T
A
M
U
A
X
A
X
T
24
23
29
33
3.5
4
3
4
101
132
99
129
C
D
E
1
2
2
1
Sample data set before invoking BIOEQ4
and changing the above information to correspond to the
desired set of data.
BIOEQ4 will rename the variables (to V ARt, VAR2,
VAR3, etc) to simplify naming conventions throughout
BIOEQ4, calculate the log.-transformed data and then
proceed to use the GLM procedure to analyze both log.transformed and untransformed data according to the
specified model.
By using the OurSTAT option in the OLM procedure,
the degrees of freedom associated with the& (estimate of
the error) can be obtained. Below is a sample of the data
produced by this option. DOSE refers to one of the byprocessing variables, LVARt shows that the analysis was
performed on the log.-transformed data of the fITSt variable
entered into the macro, and _SOURCE_ refers to the effect
entered into the model statement.
DOSE corresponds to a by-variable for processing while the
values of CMAX, 1MAX and AUcr are PK parameters
that will be on the left hand side of the model equation in
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the LS mean, and COVl and COV2 respectively correspond
to columns one and two of the variance-covariance matrix
associated with the LS mean estimates.
S
0
D
0
°
S
S
E
1
2
3
4
5
25
25
25
25
25
B
U
N
A
M
E
T
Y
P
E
R
C
E
LVARl
LVARl
LVARl
LVARl
LVARl
DAY
ERROR
SEQ
SUB (SEQ)
TREAT
D
F
S
S
The Power associated with finding a difference in
the means can be calculated by using the PROBF function
with the error degrees of freedom (d£) found in the data set
created by using the OUTSTAT option in the GLM
procedure statement, the non centrality parameter and Fstatistic as follows:
P
R
0
F
B
2 0.48 3.590.05
SS3
ERROR 24 1.39 0.19
SS3
2 0.17 0.460.63
SS3
23 8.54 6.380.01
SS3
1 0.16 1.520.23
Power=
l-Probf(Probfinv(.90,1,df,O),1,df,nc)
where
Sampleo/the data set created with the OUTSTAT option
within PROC GIM:
= log(1.2) for log.-transformed data,
= O.2*reference mean for non-transformed data.
An LSMEANS statement with COY and OUT
options is included in the GLM procedure to create a SASdata-set containing the LS means as well as the variancecovariance matrix.
L
-
S
N
D
A
B
0
S
M
E
S
E
0
1
2
3
4
S
6
7
8
9
l.0
11
l.2
l.3
14
l.5
l.6
25
25
25
25
25
25
25
25
25
25
25
25
2S
25
25
25
LVARl.
LVARl.
LVAR2
LVAR2
LVAR3
LVAR3
LVAR4
LVAR4
VAR1
VARl.
VAR2
VAR2
VAR3
VAR3
VAR4
VAR4
F
0
R
A
M
N
A
B
A
B
A
B
A
B
A
B
A
B
A
M
E
0.6
0.7
0.1
O.l.
l..1
l..l.
l..l.
l..l.
1.9
2.2
1.3
1.3
3.1
3.2
3.l.
3.3
S
T
D
E
R
·R
0.04
0.04
0.08
0.08
0.03
0.03
0.02
0.02
0.09
0.09
0.09
0.09
0.08
0.08
0.09
O.CoS
N
U
M
B
C
0
0
E
R
V
l.
V
2
0.002l.7
0.00000
0.00639
O. 0000l.
0.00089
0.00000
0.00082
0.00000
0.00933
0.0000l.
0.00863
0.0000l.
0.00796
0.0000l.
0.00802
O. 0000l.
0.00000
0.002l.7
0.00001
0.00639
0.00000
0.00089
0.00000
0.00082
0.0000l.
0.00933
0.0000l.
0.00863
0.0000l.
0.00796
0.0000l.
0.00802
l.
2
l.
2
l.
2
l.
2
1
2
l.
2
l.
2
1
2
One and two dimensional arrays are heavily used
throughout the statistical and bioequivalence sections to
refer to the statistics and variables for each of the PK
parameters. When a two dimensional array is used, it helps
distinguish transformed and non-transformed data and their
corresponding statistics.
C
The remainder of BIOEQ4 contains program code
in the form of a DATA _NULL_ to develop a simple table
containing only bioequivalence statistics. The table is
created assuming a line size of 132 and page size of 60
(usually landscape). The code can be easily modified to
create whatever table is desired since it is a typical DATA
_NULL_ data statemenL Following the SAS program code
is a more realistic copy of the table in readable format.
One data set is provided containing the necessary
information for those users who would like to customize
their tables structure and format.
Sample output from LSMEAN statement with option COY
Printed results can be obtained from the macro to
verify various steps of the data processing. These results
can be used to check the end results of the macro at various
points in the analysis, to assure the macro user that shelhe
is using BIOEQ4 correctly, and to verify the results against
previously used data processes.
DOSE is again the by-processing variable, ~AME_ gives
the names of the variables that correspond to those entered
into the macro (VARl = CMAX, VAR2 = TMAX, etc) as
well as their log.-transformed counterparts (LVARl,
LVAR2, etc), FORM denotes formulation or treatment that
defmes the bioequivalence comparitor, LSMEAN contains
BIOEQ4 can easily be developed to handle
statistical designs that have more than two formulations or
treatments present. The issue of main concern is expanding
the code to find the Call ect variance-covariance structures
out of a matrix (n x n where n is the number of formulations
or treatments) for each specific comparison to the reference.
B
A
B
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Industry Applications
trial and meet new drug application (NDA) deadlines in a
more timely manner.
Sample table created by BIOEQ4 :
nnd "", . , . II:&: KITHOUn
fI,ld •.. 1011 IUf' 111
rind
UtqtlHWlCI N . . . . . UT.,,., '''" ONIU It If
CUI
' ....1
ucr
&tIC
It
tt
II.
to
11"1 <IF mit 'fulU flHlI'll~ laa, , II)
101
.11
III
~ US
tl _stl
n . itt
... COIft_CIUln, U.II , •• un.
er tI• • •IIIIC la•• , (t)
Inn ...
loll"
*'
" • 11&
't'olIl1l1nlIlfIC .".,
.It
III
'"
... ...
.111
.tU
. t."
iii
.
....... ·'INO'II'/Il
.(1<'.11
• ("l.t,
..
."
.1•
....
• ""It" tI.I ....1 '1',LIlaIK.II. U 1a.1. ntll flllMo1',,",.......",1
tilt, S' 1I.I,",lIDTd • • .
The above figure is a bitmap file showing the general
structure and organization of the actual table BIOEQ4
creates. The full table assumes SAS fonts and is created
with a Data _Null_.
BIOEQ4 provides most required statistics from the
FDA's 1992 guidelines) as we)) as some older statistics that
were used to test average bioequivalence. It does not
provide outlier analysis, nonparametric analysis or other
analysis that may have be< 1 referenced in the 1992
guidelines.
All data and references to drug and
protocol names have been changed in this paper to protect
the confidentiality policies of the Wyeth-Ayerst Research.
Furthermore, in order to fit many of the data examples into
a two column format of this paper, numbers may have been
truncated and changed in the word processing software and
may not necessarily reflect the true nature of the data
(numeric formats may be different and variance covariance
matrices may not be symmetric, nonsensical statistical
results, etc.).
This code is continually evolving to
reflect the. changes and adaptations of Phase I
bioequivalence analysis for various governmental agencies.
In no way does the BIOEQ4 macro claim to perform all
required analysis for any agency at anyone point in time
since requirements may change. Furthermore, recognition
should be made to Stephanie Giel for her contributions in
converting the guidelines to SAS code, without which this
macro would have been extremely difficult to create.
Endnotes:
I.
2.
3.
Conclusion:
4.
Average bioequivalence anal:$'sis programs exist in
which one must hand enter the data to be used and have a
fairly simple experimental design; however, if there is a
wealth of data to be used in the analysis or if a more
complex design than a 2 by 2 crossover design is
implemented, this process can be cumbersome and difficult,
if not impossible.
5.
6.
This bioequivalence macro allows the data to be
entered into SAS in whatever manner desirable and then
allows one to specify the statistical model used in the PROC
GLM statement and returns the bioequivalence estimates, in
both data set and table format, according to the FDA's
specifications. Thus regardless of the complexity of the
statistical model or the enormity of the data, this macro
enables one to expedite the results of the bioequivalence
7.
299
FDA Office of Generic Drugs: Guidance on
statistical procedures for bioequivalence studies
using a standard two-treatment crossover design.
July 1, 1992.
Schuirmann DJ, A comparison of the two onesided tests procedure and the power approach for
of
average
assessing
the
equivalence
bioavailabillity.
J Pharmacokinetics and
Biopharmaceutics, 15: 6, 1987.
Westlake WI. Symmetrical confidence interval for
bioequivalence trials. Biometrics 322: 741-744,
1976.
Westlake WI. Statistical aspects of comparative
bioavailability trials. Biometrics 35: 273-280,
1979.
Westlake WJ. Design and statistical evaluation of
bioequivalence studies in man. In Blanchard I.
Sawchuk RI, Brodie BB, ed. Principles and
Perspectives in Drug Bioavailability, 1979, Karger
S,NewYork.
Locke CS. An exact confidence interval from
untransformed data for the ratio of two
formu1ation means. I Pharmacokinetics and
Biopharmaceutics. 12:6, 1984.
Midha KK, Ormsby ED.
Logarithmic
transformation in bioequivalence: application with
two formulations of perphenazine.
J
Pharmacetuical Sciences. 82: 2, 1993.
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TITLE1 ABOUT DRUG xxx PROTOCOL YYY
TITL2 ... MORE ABOUT YYY
TITLE3
TITLE4
BIOEQUIVALENCE PARAMETER ESTIMATES WHERE DOSE IS 17.5 MG
AUCT
AUC
76
96
97
93
101
101
92 - 111
92- 110
101
101
CMAX
TMAX
79
109
POWER BASED ON LOG-TRANSFORMED DATA(t)
RATIO OF LEAST SQUARES GEOMETRIC MEANS * (t)
90t CONFIDENCE LIMITS AROUND THE RATIO
OF THE GEOMETRIC MEANS * l')
93 - 121
90 - 123
RATIO OF LEAST SQUARE ARITHMETIC MEANS
107
105
.001
.04
.04
.001
TWO ONE-SIDED TESTS:
P(R<0.8)
P (R>l. 2)
* BASED ON THE MEAN SQUARE ERROR AND LS MEANS FROM THE LOG-TRANSFORMED ANOVA
THIS IS WHERE THE FOOTNOTE GOES
Sample BIOEQ4 table
'.3.4.,.,.1
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.001
.001
.001
.001